Specific Heat Capacity varies as temperature vary?

I remember that when I was studying in Secondary School, Physics taught me the formula which is E=mcΔT. It implies that the amount of energy required to increase an object is directly proportional to its increase of temperature, which seems very reasonable to me. Since I suppose the implicit assumption in Physics is that there is not net different in terms of energy between the increase of temperature from x°C to x+1°C and increase of temperature from y°C to y+1°C when x is not equal to y. An inherent uniformity of nature across the temperature is thus assumed.
So you can imagine my shock when I later learn that in Advance Physics there is a concept know as specific heat capacity at a specific temperature, i.e. it implies there is a difference in terms of energy between the increase of temperature from x°C to x+1°C and increase of temperature from y°C to y+1°C when x is not equal to y. Thus, uniformity of nature across the temperature is just an assumption by naive Physicist. Since energy reflect the internal state of a material, thus the nature of an iron in 5°C compare to an iron in 10°C is different. How different and what is the difference? How to account for this difference?